Transverse mass information

The transverse mass is a useful quantity to define for use in particle physics as it is invariant under Lorentz boost along the z direction. In natural units, it is: $${\displaystyle m_{T}^{2}=m^{2}+p_{x}^{2}+p_{y}^{2}=E^{2}-p_{z}^{2}}$$

• where the z-direction is along the beam pipe and so
• ${\displaystyle p_{x}}$ and ${\displaystyle p_{y}}$ are the momentum perpendicular to the beam pipe and
• ${\displaystyle m}$ is the (invariant) mass.

This definition of the transverse mass is used in conjunction with the definition of the (directed) transverse energy $${\displaystyle {\vec {E}}_{T}=E{\frac {{\vec {p}}_{T}}{|{\vec {p}}|}}={\frac {E}{\sqrt {E^{2}-m^{2}}}}{\vec {p}}_{T}}$$ with the transverse momentum vector ${\displaystyle {\vec {p}}_{T}=(p_{x},p_{y})}$. It is easy to see that for vanishing mass (${\displaystyle m=0}$) the three quantities are the same: ${\displaystyle E_{T}=p_{T}=m_{T}}$. The transverse mass is used together with the rapidity, transverse momentum and polar angle in the parameterization of the four-momentum of a single particle: $${\displaystyle (E,p_{x},p_{y},p_{z})=(m_{T}\cosh y,\ p_{T}\cos \phi ,\ p_{T}\sin \phi ,\ m_{T}\sinh y)}$$

Using these definitions (in particular for ${\displaystyle E_{T}}$) gives for the mass of a two particle system:

${\displaystyle M_{ab}^{2}=(p_{a}+p_{b})^{2}=p_{a}^{2}+p_{b}^{2}+2p_{a}p_{b}=m_{a}^{2}+m_{b}^{2}+2(E_{a}E_{b}-{\vec {p}}_{a}\cdot {\vec {p}}_{b})}$

${\displaystyle M_{ab}^{2}=m_{a}^{2}+m_{b}^{2}+2\left(E_{T,a}{\frac {\sqrt {p_{a,x}^{2}+p_{a,y}^{2}+p_{a,z}^{2}}}{p_{T,a}}}E_{T,b}{\frac {\sqrt {p_{b,x}^{2}+p_{b,y}^{2}+p_{b,z}^{2}}}{p_{T,b}}}-{\vec {p}}_{T,a}\cdot {\vec {p}}_{T,b}-p_{z,a}p_{z,b}\right)}$

${\displaystyle M_{ab}^{2}=m_{a}^{2}+m_{b}^{2}+2\left(E_{T,a}E_{T,b}{\sqrt {1+p_{a,z}^{2}/p_{T,a}^{2}}}{\sqrt {1+p_{b,z}^{2}/p_{T,b}^{2}}}-{\vec {p}}_{T,a}\cdot {\vec {p}}_{T,b}-p_{z,a}p_{z,b}\right)}$

Looking at the transverse projection of this system (by setting ${\displaystyle p_{a,z}=p_{b,z}=0}$) gives:

${\displaystyle (M_{ab}^{2})_{T}=m_{a}^{2}+m_{b}^{2}+2\left(E_{T,a}E_{T,b}-{\vec {p}}_{T,a}\cdot {\vec {p}}_{T,b}\right)}$

These are also the definitions that are used by the software package ROOT, which is commonly used in high energy physics.